The harmonic progression is 1/a, 1/(a+d), 1/(a+2d), ..., 1/(a+(n-1)d), ... These are the reciprocals of the terms of an arithmetic progression, which is the normal mathematical definition of a valid harmonic progression.
8th term is 1/(a+7d)=1/2 and 14th term is 1/(a+13d)=1/4. Therefore a+7d=2 and a+13d=4.
Subtracting we get 6d=2, d=⅓ and a=2-7/3=-⅓. Now we have a problem, because the second term would be 1/(-⅓+⅓)=1/0, which is infinitely large. It would therefore appear that this is not a mathematically true (valid) harmonic progression.
Ignoring this for the moment, the 20th term would be 1/(-⅓+19/3)=1/(18/3)=⅙.
It is possible that "harmonic progression" has different mathematical meanings, which would explain perhaps why the given progression appears to have an anomaly. I have used what I believe to be the accepted valid definition.
To avoid the anomaly, the progression would need to start at the third term 1/(a+2d)=1/(-⅓+⅔)=3, and the progression would be 3, 3/2, 1, 3/4, 3/5, 1/2, 3/7, 3/8, 1/3, 3/10, 3/11, 1/4, ... so the 6th and 12th terms (rather than the 8th and 14th terms) would be ½ and ¼, a valid harmonic progression.